You don't want, and won't need power. The station should be solar powered with battery backup. If a station were connected to power there would be a much higher risk of damage from lightening and surges.
Stations don't mount by sitting "on" something. They typically mount by attaching the north facing side to a post or mast - the south facing side will typically have a solar panel so the mount is on the north side so it won't cast a shadow.
You typically want to mount the temp/hum sensors 1 meter (3-4 foot) over the ground cover that is typical of the region - that's an okay location for the rain gauge as well. What you'd like to avoid is a brick pedestal that will adsorb and radiate solar energy differently than
the ground cover that is typical of the region. A metal mast or wooden post would be best. The less mass it has the less solar energy it can
adsorb and store, and the color, or more specifically, LRV (Light Reflectance Value) will determine how much solar energy it absorbs vs. reflects.
I would avoid station which use Wifi or Bluetooth. While those may be useful to connect a console to the internet or a phone/ tablet to the console, the station should send data to it's console using 400 or 900 Mhz band burst mode spread spectrum radios - they're super low power, good range, low interference.
You won't be able to get useful wind data at that height. The wind instruments should be located at 10 meters (33 feet), assuming the station allows you to separate them. You might consider forgoing wind measurement, which could save money, if you can't get the instruments in the right place. There are about 10 stations within a few miles of mine, and the wind data quality from them is ridiculously variable. Only about 20% produce valid wind data, the other 80% clearly weren't sited correctly, they're just there. The poorly sited stations tend to report wind speeds around 33% of actual, and the direction data is just erratic. The software also contributes to the wind data problem. Wind Speed should be a 2-minute average and Wind Gust should be 10-minute peak. One of my neighbors has them swapped - the Speed is always higher than the Gust. Others report Gust as exactly the same value as Speed. Still others don't look the the software is doing the correct sampling rate and averaging. Wind is one of the most difficult (read expensive) parameters to measure correctly. You won't be able to get accuracy in your price range, and even if money were no object you'd need to provide a mounting location that is 10 meters Above Ground Level (AGL). You will, however, get wind instruments that are permanently attached to an all-in-one station in your price range - frankly, if I were on a budget I'd prefer to skip the pointless instruments and put more money into the others, but manufacturers mostly don't share my preference.
Barometric pressure is the easiest. The pressure sensor is typically indoors in the console. Pressure gets tricky because the sensor measures "Station Pressure", which 100% of the time gets corrected to something else, like Mean Seal Level Pressure (MSLP) or Altimeter - MSLP is also corrected to temp/hum whereas Altimeter is only corrected for elevation. Without going down this rabbit hole, lets just say that pressure accuracy can become clouded by apples-to-oranges comparisons.
You should be able to get reasonable temp/hum in your price range. Accurate temperature is more dependent on shielding than height of the location. Heat energy can transfer through conduction, convection, and radiation. To accurately measure the air temperature you need to reject heat energy that arrives via radiation, and that's where radiation shields come in. There is always going to be some error as a result to radiation no matter what you spend. It should go without saying that the more you spend the better the shielding and the lower the error - there really isn't bogus vs. non-bogus, it's more a matter of "acceptable error percentage", and unfortunately there really isn't any good study data in the lower price ranges. This study
https://cdn.shopify.com/s/files/1/0515/5992/3873/files/AN_24-temp-radiation-shield-comparison.pdf while more then 20 years old is still quite relevant. While these shields alone would represent your entire budget, if not many times more, it gives you an idea of what you might expect as nothing that you can afford will be able to match the performance of these shields, and the passive shields in general have some pretty large errors. One key to accuracy from a radiation shield is active, fan aspiration, and that's not in your price range. Consequently, you're probably looking at a price range that has an average 10 degree error and only a few degrees difference between them. Does it matter to you that one station is 1 degree more accurate then another if it still has a 9 degree error?